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 A090319 Fifth column (k=4) of triangle A084938. 2
 1, 4, 14, 52, 217, 1040, 5768, 36992, 272584, 2285184, 21550656, 226071744, 2611146384, 32911082496, 449243785728, 6598780563456, 103734755882496, 1737181702840320, 30866291090657280, 579859321408266240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..150 FORMULA a(n) = Sum_{k=0..n} A090595(k)*(n-k)!. a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!. a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k). G.f.: (Sum_{k>=0} k!*x^k)^4. From G. C. Greubel, Dec 29 2019: (Start) a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function. a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End) MAPLE seq( (n+3)!*add(add(add( Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019 MATHEMATICA Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k, 0, n}, {m, 0, k}, {j, 0, m}], {n, 0, 20}] (* G. C. Greubel, Dec 29 2019 *) PROG (PARI) vector(21, n, my(b=binomial); sum(k=0, n-1, sum(m=0, k, sum(j=0, m, (n-1)!/(b(n-1, k)*b(k, m)*b(m, j)) )))) \\ G. C. Greubel, Dec 29 2019 (Magma) F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n, k)*B(k, m)*B(m, j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019 (Sage) b=binomial; [sum(sum(sum(factorial(n)/(b(n, k)*b(k, m)*b(m, j)) for j in (0..m)) for m in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019 (GAP) B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], m-> Sum([0..m], j-> Factorial(n)/(B(n, k)*B(k, m)*B(m, j)) )))); # G. C. Greubel, Dec 29 2019 CROSSREFS Cf. A084938. Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595. Sequence in context: A308023 A149489 A125783 * A295518 A047118 A017948 Adjacent sequences: A090316 A090317 A090318 * A090320 A090321 A090322 KEYWORD easy,nonn AUTHOR Philippe Deléham, Feb 05 2004 STATUS approved

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Last modified June 7 11:12 EDT 2023. Contains 363157 sequences. (Running on oeis4.)